L(s) = 1 | − 3.91·5-s + 1.60·7-s + 5.85·11-s + 0.924·13-s − 6.27·17-s − 2.72·19-s − 0.550·23-s + 10.3·25-s − 0.556·29-s + 9.36·31-s − 6.29·35-s + 6.50·37-s − 2.46·41-s − 2.43·43-s − 4.96·47-s − 4.40·49-s + 3.80·53-s − 22.8·55-s + 10.0·59-s − 9.11·61-s − 3.61·65-s − 2.78·67-s − 12.9·71-s + 15.6·73-s + 9.41·77-s + 1.12·79-s − 16.5·83-s + ⋯ |
L(s) = 1 | − 1.74·5-s + 0.608·7-s + 1.76·11-s + 0.256·13-s − 1.52·17-s − 0.625·19-s − 0.114·23-s + 2.06·25-s − 0.103·29-s + 1.68·31-s − 1.06·35-s + 1.06·37-s − 0.384·41-s − 0.371·43-s − 0.724·47-s − 0.629·49-s + 0.523·53-s − 3.08·55-s + 1.30·59-s − 1.16·61-s − 0.448·65-s − 0.340·67-s − 1.54·71-s + 1.83·73-s + 1.07·77-s + 0.126·79-s − 1.81·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.419950045\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.419950045\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 - T \) |
good | 5 | \( 1 + 3.91T + 5T^{2} \) |
| 7 | \( 1 - 1.60T + 7T^{2} \) |
| 11 | \( 1 - 5.85T + 11T^{2} \) |
| 13 | \( 1 - 0.924T + 13T^{2} \) |
| 17 | \( 1 + 6.27T + 17T^{2} \) |
| 19 | \( 1 + 2.72T + 19T^{2} \) |
| 23 | \( 1 + 0.550T + 23T^{2} \) |
| 29 | \( 1 + 0.556T + 29T^{2} \) |
| 31 | \( 1 - 9.36T + 31T^{2} \) |
| 37 | \( 1 - 6.50T + 37T^{2} \) |
| 41 | \( 1 + 2.46T + 41T^{2} \) |
| 43 | \( 1 + 2.43T + 43T^{2} \) |
| 47 | \( 1 + 4.96T + 47T^{2} \) |
| 53 | \( 1 - 3.80T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 + 9.11T + 61T^{2} \) |
| 67 | \( 1 + 2.78T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 - 1.12T + 79T^{2} \) |
| 83 | \( 1 + 16.5T + 83T^{2} \) |
| 89 | \( 1 - 7.96T + 89T^{2} \) |
| 97 | \( 1 + 4.68T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.230532882148054433059932314807, −7.39677084973947636849604694004, −6.65291359948800113227687169664, −6.27319972223877666360981627066, −4.82556788884996335169594672030, −4.30724959064858760728687826068, −3.91783720235151907917051842959, −2.94809382749922121961845169926, −1.71943554521531121787282913663, −0.64410373298850437470337297996,
0.64410373298850437470337297996, 1.71943554521531121787282913663, 2.94809382749922121961845169926, 3.91783720235151907917051842959, 4.30724959064858760728687826068, 4.82556788884996335169594672030, 6.27319972223877666360981627066, 6.65291359948800113227687169664, 7.39677084973947636849604694004, 8.230532882148054433059932314807